Internal problem ID [5677]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear
Equations: Ordinary Points. Page 169
Problem number: 1(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime }-x^{2} y-1=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 72
Order:=8; dsolve(diff(y(x),x$2)+diff(y(x),x)-x^2*y(x)=1,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}-\frac {1}{2520} x^{7}\right ) y \relax (0)+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {7}{120} x^{5}-\frac {19}{720} x^{6}+\frac {13}{1680} x^{7}\right ) D\relax (y )\relax (0)+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+\frac {13 x^{6}}{720}-\frac {11 x^{7}}{1680}+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 126
AsymptoticDSolveValue[y''[x]+y'[x]+x^2*y[x]==1,y[x],{x,0,7}]
\[ y(x)\to \frac {31 x^7}{5040}-\frac {11 x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}+c_1 \left (\frac {x^7}{2520}-\frac {x^6}{360}+\frac {x^5}{60}-\frac {x^4}{12}+1\right )+c_2 \left (-\frac {37 x^7}{5040}+\frac {17 x^6}{720}-\frac {x^5}{24}-\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]